Gelfreich, Vassili and Gelfreikh, Natalia. (2009) Unique resonant normal forms for area-preserving maps at an elliptic fixed point. Nonlinearity, Vol.22 (No.4). pp. 783-810. ISSN 0951-7715

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Abstract

We study possible simplifications of normal forms for area-preserving maps near a resonant elliptic fixed point. In the generic case we prove that at all orders the Takens normal form vector field can be transformed to the particularly simple form described by the formal interpolating Hamiltonian H = I-2 A(I) + I-n/2 B(I) cos n phi. The form of formal series A and B depends on the order of the resonance n. For each n >= 3 we establish which terms of the series can be eliminated by a canonical substitution and derive a unique normal form which provides a full set of formal invariants with respect to canonical changes in coordinates. We extend these results to families of area-preserving maps. Then the formal interpolating Hamiltonian takes a form similar to the case of an individual map but involves formal power series in action I and the parameter.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Nonlinearity
Publisher:	Institute of Physics Publishing Ltd.
ISSN:	0951-7715
Date:	April 2009
Volume:	Vol.22
Number:	No.4
Number of Pages:	28
Page Range:	pp. 783-810
Identification Number:	10.1088/0951-7715/22/4/006
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Royal Society (Great Britain)
URI:	http://wrap.warwick.ac.uk/id/eprint/28325

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